due to frictionless contact. Non-linear geometry is also taken into account in order to simulate large deformations proper of collapse issue. The dynamic analysis developed, is justified by the fact that the ratio between kinetic energy (ALLKE) and strain energy (ALLSE) for the whole model is kept below 0.1 until the collapse point, in Figure 2.8 kinetic and strain energies are plotted against the dimensionless load.
In this model, C3D8R (eight-node continuum linear brick elements with reduced integration and hourglass control) is used for the three parts as can be seen in Figure 2.9. This type of components matches very good the purpose of the model, in fact these can be used for linear and complex non-linear analysis producing high accuracy results when contacts, plasticity and non-linear geometry are considered [8].
Figure 2.8 ALLKE/ALLSE versus dimensionless load.
Figure 2.9 Carcass model mesh details, cross-section, and surfaces.
2.3 Comparison and Discussion
The input data for the design case of a flexible pipeline is the internal diameter which is the same for both steel strip reinforced thermoplastic pipes and the case improved against external pressure with stainless steel carcass. For both numerical and numerical models, the chosen reference surface is the outermost and both displacement along x and y directions are analyzed. For the numerical simulation, the central body is chosen as reference, being the only full profile in the model. For the numerical case, displacements could differ one to the other because it simulates the actual geometry as it is shown in Figures 2.10 and 2.11 While for some theoretical limits, they will be coincident in both directions. By simulating the mechanical behavior of the structure, it is possible to catch the buckling pressure.
Firstly, using the parameters listed in Eq. (2.14) for Eq. (2.10), it is possible to carry out the smallest moment of inertia of the cross-section: I2’ = 185. 41 mm4. Then, from Eq. (2.9), the equivalent stiffness per unit length is calculated EIeq = 2,317,671.93 MPa mm3 if the mean radius is computed as follows: R = Rinn + t = 79.40 mm. Finally, the critical buckling pressure for a ring without ovality is obtained and it is equal to: pcr = 13.89 MPa. This result is normalized by the critical load at L = 0.04 as shown in Eq. (2.11), and it is plotted in Figure 2.14, which shows the sudden collapse when the structure reaches the collapse pressure.
When the imperfection is considered, the initial value of ovalization taken into account is1 = 0. 002, as minimum requirement, so that the initial displacement can be computed using Eq. (2.3), and it is equal to uR1 = 0.1652, referring the computation to the external radius. The outcomes of the theoretical model can be finally plotted, as shown in Figure 2.14, against the dimensionless load and for the ovality computed for each load increment as follows:
where, Dmax and Dmin are the maximum and minimum diameters uploaded step by step from the results of the of the displacements, along x and y directions, respectively.
In the same way, outcomes for finite element model are computed extracting U1 and U2 and they are plotted in Figure 2.12. As previously said, it can be seen that the external surface displays different magnitude of displacements along x and y directions, as shown in Figures 2.10 and 2.11.
As it was shown by Gay [7], the pre-buckling behaviors match one to each other with very low error. In Figure 2.14, it can be observed that for small value of external pressure the ovality increases linearly. In this phase, the very small difference between theoretical and numerical models can be caused by the fact that the theoretical model regards as an equivalent ring. Vice versa, the actual geometry shows gaps among the different parts, so it is obvious that results show a slightly different about stiffness. When the pressure increases, a non-linear trend is illustrated for both models. Numerical results show wider ovality for the same load, compared to analytical outcomes because of theoretical limits that consider as critical load the one for the case with no imperfections. As it was expected, the comparison between results makes sense in order to understand the pre-buckling behavior in terms of displacements and not in terms of collapse load. In fact, from Figure 2.12, the relevant output that comes out, in terms of pressure, is an error equal to 36% between theoretical and numerical outcomes when dealing with imperfections Table 2.3.
Figure 2.10 U1 displacements.
Figure 2.11 U2 displacements.
Figure 2.12 Ovality versus dimensionless load.
Table 2.3 Collapse pressures.
| Model | Collapse pressure [MPa] |
|---|---|
| No imperfections | 13.89 |
| Theoretical | 13.19 |
| FEM | 8.43 |
Moreover, here, it can be confirmed that, for a value of ovality equal to 4%, the external pressure no longer increases, while the ovality keep to sharply rise,
